Topics in Mathematical Biology and Fluid Mechanics

该项目侧重于数学生物学和流体力学的几个方面。许多重要的生物现象涉及流体-细胞相互作用或流体样行为。例如，海洋动物卵的受精依赖于周围流体流的运输，沙丁鱼形成大的流体状群体以躲避捕食者。该项目的主要重点是开发新的数学工具来分析这些生物现象中各种与流体相关或类似流体的效应。这个项目有三个主要主题。第一个主题是关于流体流动与生物现象之间的相互作用，这是由海洋动物受精和细胞胚胎学的两个实验激发的。在第一个实验中，研究人员观察到最佳的流体速度，以最大限度地提高鲍鱼的受精率。第二个实验证实了细胞流有助于果蝇胚胎早期器官的形成。流体混合效应和快速扩散效应在分析中起着重要作用。这两种效应在生物系统中引入了另一种“快速扩散时间尺度”，并显著改变了生物系统的长期行为。首席研究员（PI）计划开发新的数学工具，以捕捉这些流体现象和所涉及的生物学之间的相互作用。 该项目将为本科生提供研究培训经验。J. Bedrossian和PI在论文中应用的修正的hypoprovity泛函和详细的谱分析是推导非线性设置下“快速扩散时标”的关键。将这些工具应用于生物学相关模型提供了对实验的更深入理解。第二个主题涵盖了群集行为的单种群和多种群流体动力学模型。在这些模型中，群体的代理收敛到一个有限的结构状态，通过个人之间共享信息。然而，如果代理人只与他们的直接邻居共享信息，羊群的限制行为是非常不平凡的。PI计划开发新的工具，并从谱图理论和随机图理论的思想来分析它。第三个主题是关于流体力学中的增长/爆破机制。PI将研究两个非局部模型，这些模型与由3D-Euler方程的奇异Hou-Luo情景激发的流体中小尺度爆破的产生有关。这个奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project focuses on several aspects of mathematical biology and fluid mechanics. Many critical biological phenomena involve fluid-cell interaction or fluid-like behaviors. For example, fertilization of marine animal eggs relies on the transportation of the ambient fluid stream and sardines form large fluid-like swarms to evade predators. The project's principal focus is to develop novel mathematical tools to analyze the various fluid-related or fluid-like effects in these biological phenomena. There are three main topics in this project. The first topic concerns the interaction between the fluid flow and biological phenomena, which is motivated by two experiments in marine animal fertilization and cell embryology. In the first experiment researchers observed optimal fluid speed to maximize the fertilization rate of abalone. The second experiment confirms that the cellular flow helps organ formation at the early stages of the embryos of Drosophila. Fluid mixing effect and fast-spreading effect play important roles in the analysis. These two effects introduce another 'fast diffusion time scale' into the biological system and significantly change the long-time behavior. The principal investigator (PI) plans to develop new mathematical tools that capture the interplay between these fluid phenomena and the biology involved. The project will provide research training experience for undergraduate students.A modified hypocoercivity functional, which was applied in the paper by J. Bedrossian and the PI, and detailed spectral analysis, are crucial to derive the 'fast diffusion time scale' in the nonlinear setting. Applying these tools to the biologically relevant models provides deeper understandings of the experiments. The second topic covers the single and multi-species hydrodynamic models of flocking behavior. In these models, the flock of agents converges to a limiting structural state through sharing information among individuals. However, if the agents only share information with their direct neighbors, the flock's limiting behavior is highly nontrivial. The PI plans to develop new tools and bring ideas from spectral graph theory and random graph theory to analyze it. The third topic concerns the growth/blow-up mechanism in fluid mechanics. The PI will study two non-local models related to the generation of small-scale blowup in the fluid motivated by the singular Hou-Luo scenario for the 3D-Euler equation. The tools developed here may turn out to be useful in some biological models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.